IsLittleOPower

formal source

  27  IsBigO f (fun x => x ^ n) 0
  28
  29/-- f = o(x^n) as x → 0. -/
  30def IsLittleOPower (f : ℝ → ℝ) (n : ℕ) : Prop :=
  31  IsLittleO f (fun x => x ^ n) 0
  32
  33/-- Constant function is O(1). -/
  34theorem const_is_O_one (c : ℝ) :
  35  IsBigO (fun _ => c) (fun _ => 1) 0 := by
  36  unfold IsBigO
  37  have hpos : (0 : ℝ) < |c| + 1 := by have := abs_nonneg c; linarith
  38  refine ⟨|c| + 1, hpos, 1, by norm_num, ?_⟩
  39  intro x _
  40  have h1 : |c| ≤ |c| + 1 := by linarith
  41  simp only [abs_one, mul_one]
  42  exact h1
  43
  44/-- Linear function is O(x). -/
  45theorem linear_is_O_x (c : ℝ) :
  46  IsBigO (fun x => c * x) (fun x => x) 0 := by
  47  unfold IsBigO
  48  have hpos : (0 : ℝ) < |c| + 1 := by have := abs_nonneg c; linarith
  49  refine ⟨|c| + 1, hpos, 1, by norm_num, ?_⟩
  50  intro x _
  51  rw [abs_mul]
  52  have h1 : |c| ≤ |c| + 1 := by linarith
  53  have h2 : |c| * |x| ≤ (|c| + 1) * |x| := by
  54    apply mul_le_mul_of_nonneg_right h1 (abs_nonneg _)
  55  exact h2
  56
  57/-- x² is O(x²) (reflexive). -/
  58theorem x_squared_is_O_x_squared :
  59  IsBigOPower (fun x => x ^ 2) 2 := by
  60  unfold IsBigOPower IsBigO